Advanced High-School Mathematics

280 CHAPTER 5 Series and Differential Equations

(See the footnote.^16 ) By thinking in terms of binary decimal

representations (See, e.g., Exercise 4 on page 92) argue that

any real numberywith|y|≤2 can be represented in the form

Σ+.

(b) If Σ+=y, show that Σ+−Σ−= 2(y−1).

(c) Conclude that any numberxbetween−2 and 2 can be repre-

sented as Σ+−Σ−and hence as the infinite series

∑∞

n=0

n

2 n

. (This
result is not true if the series is replaced by, say, one of the
form

∑∞

n=0

n

3 n

. While the values of such a series would always
lie between−^32 and^32 , and despite the fact that uncountably
many such numbers would occur, the set of such numbers is
still very small.^17

5.2.4 The Dirichlet test for convergence (optional discussion)

There is a very convenient test which can be thought of as a gener-
alization of the alternating series test and often applies very nicely to
testing for conditional convergence. For example, we may wish to test

the series

∑∞

n=1

cosn

n

for convergence. It is not clear whether this series

is absolutely convergent^18 , nor is it an alternating series, so none of the
methods presented thus far apply.

Let us first consider the following very useful lemma.

Lemma. Let(an)and(bn)be two sequences and setsn=

∑n

k=1

ak. Then

one has

(^16) There is an important result being used here, namely that if∑unis an absolutely convergent
series, then its sum is unaffected by any rearrangement of its terms. 17
The smallness of this set is best expressed by saying that it has “measure zero.” Alternatively, if
we were to select a real number randomly from the interval[−^32 ,^32 ], then the probability of selecting
a number of the form
∑∞
n=0
n
3 nis zero. Finally, if instead of allowing the numeratorsnto be±1 we
insisted that they be either 0 or 2, then what results is the so-calledCantor Ternary Set(which
also has measure zero). 18
It’s not, but this takes some work to show.